Many solids and their bounding surfaces are generated by taking a 2-d
figure and rotating it about an axis, resulting in a circular
cross-section. Examples include spheres from circles, spheroids from
ellipses, cones from triangles, bicones and cylinders from squares, and
many more complex forms.
But the circle isn't the only curve of constant width, so what if you
trace out a surface by rotating a 2-d figure about a non-circular curve
of constant width?
The simplest example might be the reuleaux triangular cylinder, but one
could imagine reuleaux triangle versions of other surfaces of
revvvolution.
Let's take an equilateral triangle whose edge length is the width of a
reuleaux triangle, place it so it's base is in the reuleaux triangle,
and the equilateral triangle is in a plane normal to the plane of the
reuleaux triangle and rotate the equilateral triangle so its base
vertices trace the reuleaux triangle. Alternatively, let's hold one
endpoint of a segment at a fixed point above the center of the reuleaux
triangle and move the other endpoint of the segment to trace the
reuleaux triangle, letting the segment trace out a ruled surface. Are
these constructions the same? I suspect not and that the apex of the
equilateral triangle moves in the process,, which raises the question of
what path the apex traces, which I believe to be equivalent to the
question of what path the midpoint of a maximally long segment traces as
it rotates within a reuleaux triangle. If this intuition is correct,
things like a reuleaux sphere or speroid would have the same issue of a
shifting apex.
I also find myself curious what shape would result if we took a segment,
held one end static above the center of a deltoid or asteroid, and
traced the deltoid/asteroid with the other end of the segment... Or
placed a deltoid and reuleaux triangle with matching vertices in
parallel planes and trace them with opposite ends of a segment... does
the resulting surface have a cross section that is a equilateral
triangle? And is there a logical means of extending such a shape further
beyond either the deltoid or reuleaux triangle? If so, my intuition is
that there's a point where the "pinched" end becomes just three segments
forming a Y-shape with 3 120-degree angles and at the other end becomes
a circle.
Is there a name for the class of ruled surfaces generated by holding one
end of a segment at a fixed point(the apex) while the other end traces
an arbitrary plane curve? If not, I'm tempted to call them conoids, but
a quick google search suggests that term is already used for a different
class of surfaces.